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Decaying positive solutions to some quasilinear differential equations. (English) Zbl 0944.34005
The author deals with the existence of decaying positive solutions \((u,\lambda)\) \((u\in C^1([0,\infty))\), \(\lambda \in (0,\infty)\)) to the two equations \[ \begin{aligned} D_au+\lambda r^a F(r,u,|u'|)&=0, \qquad r>0,\\ D_au+\lambda r^a F_0(r,u)&=0, \qquad r>0, \end{aligned} \] with \(D_au = (r^a |u'|^{p-2}u')'\), \(a>1\), \(p\in (1,1+a)\). The existence of solutions to the equations for the subhomogeneous case (\(F(r,tU,t|U'|)/t^{p-1}\to 0\) as \(t\to \infty \)) enables to prove existence theorems for more general cases using the super-sub-solution method.
34A34 Nonlinear ordinary differential equations and systems
35J70 Degenerate elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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